Classical orthogonal polynomials in two variables: a matrix approach (Q1776174)

In the past, several authors have tried to establish a characterization of two-variable orthogonal polynomials along the line of the Bochner-classification of classical orthogonal polynomials in 1929. In 1967, \textit{H. L. Krall} and \textit{I. M. Sheffer} [Ann. Mat Pura Appl., IV. Ser. 76, 325--376 (1967; Zbl 0186.38602)] extended the Bochner approach to the study of a second order partial differential equation for the weight function \(w(x,y)\) with polynomial coefficients of certain degrees in \(x\) and \(y\), but that classification did not include for instance the tensor product of the Jacobi polynomials in one variable as these did not satisfy a differential equation of the type studied. The authors of this paper, however, succeeded in giving an extension of the just mentioned results that does include tensor products of one variable polynomials. The main tools are 2D moment functionals taken from the topological dual of the space of polynomials in two variables and the calculus for these functionals (including distributional counterparts to the gradient and divergence operators, originally defined on the polynomials). An important role is played by the matrix equation \[ L[\mathbf{P}_n^T]=\text\textbf{P}_n^T\Lambda_n^T\;(n\geq 0), \] where \(L[p]=\text{div}(\phi\nabla p)+{\tilde{\psi}}\nabla p\) and (\textbf{P}\(_n)_n\) is the corresponding weak orthogonal polynomial system. A quite interesting paper.